Limiting distributions of the classical error terms of prime number theory

TL;DR

This paper proves a general theorem demonstrating the existence of limiting distributions for various error terms in prime number theory, unifying and extending previous results across multiple related problems.

Contribution

It introduces a broad theorem that establishes limiting distributions for a wide class of prime number error terms, including new cases like automorphic L-functions and Chebotarev's density theorem.

Findings

Established limiting distributions for prime number theorem error terms.

Unified previous results by Wintner, Rubinstein, Sarnak, and Ng.

Extended to automorphic L-functions and Chebotarev's theorem error terms.

Abstract

In this article we prove a general theorem which establishes the existence of limiting distributions for a wide class of error terms from prime number theory. As a corollary to our main theorem, we deduce previous results of Wintner (1935), Rubinstein and Sarnak (1994), and of Ng (2004). In addition, we establish limiting distribution results for the error term in the prime number theorem for an automorphic $L$-function, weighted sums of the M\"{o}bius function, weighted sums of the Liouville function, the sum of the M\"{o}bius function in an arithmetic progression, and the error term in Chebotarev's density theorem.